CN110045323B - Matrix filling-based co-prime matrix robust adaptive beamforming algorithm - Google Patents

Matrix filling-based co-prime matrix robust adaptive beamforming algorithm Download PDF

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CN110045323B
CN110045323B CN201910194101.XA CN201910194101A CN110045323B CN 110045323 B CN110045323 B CN 110045323B CN 201910194101 A CN201910194101 A CN 201910194101A CN 110045323 B CN110045323 B CN 110045323B
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CN110045323A (en
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杨桐
郑植
王文钦
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University of Electronic Science and Technology of China
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    • G—PHYSICS
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    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/14—Systems for determining direction or deviation from predetermined direction
    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
    • G—PHYSICS
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    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
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    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
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Abstract

The invention provides a matrix filling-based co-prime matrix robust adaptive beamforming algorithm, which comprises the following steps: calculating a sample covariance matrix of the received data; vectorizing the sample covariance matrix to obtain a vector, and then performing redundancy removal and vector rearrangement on the vector to obtain a received data vector of a complete co-prime matrix differential optimization array; filling 0 in all the elements with discontinuous wave path difference in the received data vector to obtain a vector, and then taking the information of the positive half part of the vector to obtain the vector; expanding the vector into a Topritz matrix; restoring the Topritz matrix to obtain a filling covariance matrix; carrying out spectral peak search in an interference signal angle region to obtain estimation of an arrival angle of each interference signal; utilizing the estimated arrival angle of the interference signal, the physical array information of the co-prime array and reconstructing an interference and noise covariance matrix of the co-prime array physical array; the weight vectors of the adaptive beamformer are calculated using the interference-plus-noise covariance matrix and the estimate of the desired signal steering vector.

Description

Matrix filling-based co-prime matrix robust adaptive beamforming algorithm
Technical Field
The invention belongs to the field of array signal processing algorithms, and particularly relates to a matrix filling-based co-prime array steady adaptive beam forming algorithm.
Background
Adaptive beamforming is one of the core technologies in array signal processing, and is widely used in fields such as mobile communication, radar, medical imaging, sonar detection, and the like. The adaptive beamforming technology can adaptively change the weighting vector of the array antenna according to the received training sequence or communication signal, thereby achieving the purposes of suppressing interference and receiving the desired signal without distortion. The adaptive beam forming technology has been proved by theory and practice to be very sensitive to the problems of the error of the desired signal steering vector, the error of the covariance matrix, insufficient sampling snapshot times and the like. The factors causing these errors mainly include array element position error, inter-array element coupling, array channel amplitude-phase error, direction of arrival (DOA) estimation error of the desired signal, and local scattering. As long as the known a priori information deviates from the true values, the performance of the adaptive beamformer is severely affected. Therefore, how to reduce the sensitivity of the adaptive beam forming device to errors and improve the robustness of the adaptive beam forming algorithm is a hot spot in the technical field of beam forming at present.
In recent years, a new sparse array, a co-prime array, has been receiving attention from researchers. The co-prime array is formed by extracting some specific array elements from the uniform linear array. The array aperture is larger and the degree of freedom is more higher with fewer array elements and larger array element spacing. Compared with an even linear array, the co-prime array can obviously improve the array resolution and reduce the system cost and complexity. Compared with other types of sparse arrays, the relatively prime array has smaller array element spacing and smaller array element mutual coupling. When co-prime matrix received data is processed, a corresponding differential optimization matrix is generally obtained through high-order statistics of the co-prime matrix, so that the advantage of larger degree of freedom of a virtual array is obtained. Currently, the research on co-prime arrays is mainly focused on DOA estimation, and less research is done on adaptive beamforming. However, due to the excellent characteristics of the co-prime matrix, research on the robust adaptive beamforming technology based on the co-prime matrix is also gradually developed recently. At present, algorithms of articles of the mutually-prime matrix self-adaptive beam forming can only process receiving signals of a virtual array by utilizing an ULA part formed by a part of continuous array elements of a mutually-prime matrix differential optimization array, and the mutually-prime matrix differential optimization array has a plurality of holes, so that the algorithms cannot utilize all degrees of freedom of the mutually-prime matrix virtual array. In recent years, related researches show that a difference optimization array of a co-prime array can be interpolated by a matrix filling technology, and based on the theory, some co-prime array DOA estimation algorithms based on the interpolation technology of the difference optimization array are proposed. The core idea of the algorithms is that the holes of the discontinuous part of the differential optimization array of the co-prime array are interpolated by a matrix filling technology, so that the whole differential optimization array forms a continuous ULA, the data of all virtual array elements of the array are effectively utilized, and the degree of freedom of the differential optimization array of the co-prime array which can be actually utilized by the algorithms is improved. Similar to DOA estimation, for a robust adaptive beamforming algorithm, the larger the aperture of the array used to process the received signal and the higher the degree of freedom, the more accurately the interfering signal can be located and suppressed, resulting in better beamformer performance. Based on this fact, it can be considered to improve the performance of the robust adaptive beamforming algorithm by interpolating the difference optimized matrix of the co-prime matrix using the matrix filling technique.
Disclosure of Invention
In view of the above-mentioned shortcomings of the prior art, the present invention aims to provide a robust adaptive beamforming algorithm based on a co-prime matrix filled with matrices, which interpolates a differential optimized matrix of the co-prime matrix by using matrix filling to change the interpolation into a continuous ULA, so that more degrees of freedom can be utilized, thereby improving the output performance of an adaptive beamformer.
To achieve the above and other related objects, the present invention provides a matrix filling based co-prime array robust adaptive beamforming algorithm, the co-prime array comprising a first ULA and a second ULA, the first ULA being co-prime with the second ULA; the algorithm comprises the following steps:
computing a sample covariance matrix for received data
Figure GDA0003083536180000021
To the sample covariance matrix
Figure GDA0003083536180000022
Performing characteristic decomposition to calculate the minimum characteristic value gammamin
The sample covariance matrix
Figure GDA0003083536180000023
Vectorizing to obtain a vector z, and then performing redundancy removal and vector rearrangement on the vector z to obtain a received data vector z of the complete co-prime matrix differential optimization array1；
The received data vector z1All the elements with discontinuous medium wave path difference are filled with 0 to obtain vector
Figure GDA0003083536180000024
Then taking the vector
Figure GDA0003083536180000025
To obtain a vector
Figure GDA0003083536180000026
The vector is measured
Figure GDA0003083536180000027
Expanded into Topritz matrices Rv∈C(2MN-N+1)×(2MN-N+1)Wherein 2M is the number of array elements of the first ULA, N is the number of array elements of the second ULA, M, N are mutually prime numbers, and M is<N；
Subjecting the Topritz matrix RvRecovering to obtain a filled covariance matrix
Figure GDA0003083536180000028
For the filled covariance matrix
Figure GDA0003083536180000029
Performing feature decomposition by using the filled covariance matrix
Figure GDA00030835361800000210
Noise subspace U ofnConstructing a MUSIC spectrum; in the desired signal angle region ΘsInternally performing spectral peak search to obtain the arrival angle of the expected signal
Figure GDA00030835361800000211
Calculating a steering vector of the desired signal
Figure GDA00030835361800000212
In the angular region of the interference signal
Figure GDA00030835361800000213
The estimation of the arrival angle of each interference signal is obtained by searching the spectrum peak in the interior
Figure GDA00030835361800000214
Using the estimated arrival angle of the interference signal and the physical array information of the co-prime matrix and reconstructing the interference-plus-noise covariance matrix of the co-prime matrix physical array
Figure GDA00030835361800000215
Using the interference-plus-noise covariance matrix
Figure GDA00030835361800000216
And estimation of a desired signal steering vector
Figure GDA00030835361800000217
A weight vector w for the adaptive beamformer is calculated.
Optionally, the sample covariance matrix is calculated using K sample snapshots:
Figure GDA00030835361800000218
wherein: k is the number of sampling times, K is the serial number of sampling, x (K) is the data received by each array element of the co-prime array, xH(k) Is the conjugate transpose of x (k).
Optionally, performing redundancy removal and vector rearrangement on the vector z specifically includes:
the sums of the elements with the same path difference are averaged as calculated below
Figure GDA0003083536180000031
Wherein m represents the wave path difference of the difference optimization array, | · | represents the number of elements in the solution set,
Figure GDA0003083536180000032
representing a covariance matrix
Figure GDA0003083536180000033
Corresponding wave path difference of n1-n2The value of (a) is (b),<x>n1representing a signal vector x at a reference position n of an array element1The value of (a) is (b),<x>n2representing a signal vector x at a reference position n of an array element2The value of (d);
set T (m) represents a set of array element position doublets with wave path difference m
T(m)＝{(n1,n2)∈S2|n1-n2＝m}
Will be provided with<z1>mArranging according to the order of the wave path difference from small to large to obtain the receiving data vector z of the corresponding complete co-prime matrix differential optimization array1∈C3MN+M-N。
Optionally, said receiving said received data vector z1All the elements with discontinuous medium wave path difference are filled with 0 to obtain vector
Figure GDA0003083536180000034
Then taking the vector
Figure GDA0003083536180000035
To obtain a vector
Figure GDA0003083536180000036
The method specifically comprises the following steps:
the received data vector z1All the elements with discontinuous middle wave path difference are filled with 0, so that they form a higher-dimensional received data vector
Figure GDA0003083536180000037
Namely, it is
Figure GDA0003083536180000038
Wherein S isdiffSet of values representing positions of elements of a differentially optimised array, Sdiff＝{n1-n2|n1,n2∈S}，S＝{dnD, N is 1, 2M + N-1, d represents a half-wave long distance λ/2;
taking the received data vector
Figure GDA0003083536180000039
The positive half of the vector data is obtained to obtain a complex vector with the dimensionality of 2MN-N +1
Figure GDA00030835361800000310
Figure GDA00030835361800000311
Alternatively, a matrix filling problem is constructed to create a matrix from the Toplitz matrix RvRecovering to obtain a filled covariance matrix
Figure GDA00030835361800000312
Figure GDA0003083536180000041
Figure GDA0003083536180000042
Figure GDA0003083536180000043
n1,n2∈S+＝{n|n∈S,n≥0}
Wherein rank () represents the rank of the matrix, and the rank minimization problem is converted into the kernel norm minimization problem as follows
Figure GDA0003083536180000044
Figure GDA0003083536180000045
Figure GDA0003083536180000046
n1,n2∈S+
Wherein | · | purple sweet*Representing the kernel norm of the matrix.
Optionally, the estimation of the angle of arrival of each interference signal
Figure GDA0003083536180000047
Obtained by the following method:
for the filled covariance matrix
Figure GDA0003083536180000048
The characteristic decomposition is carried out, and the characteristic decomposition is carried out,
Figure GDA0003083536180000049
wherein, UsIs composed of
Figure GDA00030835361800000410
Signal subspace of, sigmasIs UsThe characteristic value, U, corresponding to each vectornIs composed of
Figure GDA00030835361800000411
Of noise subspace, ΣnIs UnThe characteristic value corresponding to each vector in the vector;
using the noise subspace UnConstructing a MUSIC spectrum on the whole angle area,
Figure GDA00030835361800000412
wherein θ ∈ [ -90 °,90 °]Representing angles in the search, d (θ) representing a filled covariance matrix
Figure GDA00030835361800000413
Corresponding to the steering vector at the angle θ, d (θ) < 1, ej2πdsinθ/λ,ej2π2dsinθ/λ...,ej2π(2MN-N)dsinθ/λ]T；
Assuming that the desired signal is located in a span
Figure GDA00030835361800000414
Wherein
Figure GDA00030835361800000415
Is the angle of arrival of the assumed desired signal, and the interfering signal is located at thetasBetween the supplement section
Figure GDA00030835361800000416
Performing the following steps;
in the desired signal interval thetasInternal pair MUSIC spectrum PMUSIC(theta) performing a spectral peak search, using the largest spectral peak in the region as the spectral peak of the desired signal, and determining the angular position thereof
Figure GDA00030835361800000417
Recording an estimate of the angle of arrival as a desired signal;
the estimation of the steering vector of the desired signal is as follows;
Figure GDA00030835361800000418
in the interval of interference signal
Figure GDA00030835361800000419
The arrival angle information of each interference signal is obtained by searching the spectrum peak of the MUSIC spectrum, and the estimation of the arrival angle of each interference signal is obtained after the largest (L-1) spectrum peak is selected
Figure GDA00030835361800000420
Optionally, the interference plus noise covariance matrix
Figure GDA00030835361800000421
In order to realize the purpose,
Figure GDA0003083536180000051
wherein, γminRepresenting a sample covariance matrix
Figure GDA0003083536180000052
Is determined by the minimum characteristic value of (c),
Figure GDA0003083536180000053
estimated power for the l source, I2M+N-1The dimension is a unit array of 2MN + N-1;
optionally, the weight vector w of the adaptive beamformer is:
Figure GDA0003083536180000054
as described above, the matrix filling based co-prime robust adaptive beamforming algorithm of the present invention has the following beneficial effects:
according to the invention, through a matrix filling technology, the holes of the discontinuous part of the differential optimization array of the co-prime array are interpolated, so that the whole differential optimization array forms a continuous ULA, the data of all virtual array elements of the differential optimization array are effectively utilized, and the degree of freedom of the differential optimization array of the co-prime array, which can be actually utilized by an algorithm, is improved. In the adaptive beamforming technology, the higher the degree of freedom of an array for processing a received signal is, the more accurately an interference signal can be located and suppressed, thereby obtaining better beamforming performance.
Drawings
To further illustrate the description of the present invention, the following detailed description of the embodiments of the present invention is provided with reference to the accompanying drawings. It is appreciated that these drawings are merely exemplary and are not to be considered limiting of the scope of the invention.
FIG. 1 is a co-prime array structure;
fig. 2 is a plot of output SINR versus input SNR for the tested beamformers;
FIG. 3 is a plot of output SINR of the tested beamformer as a function of sample snapshot number;
fig. 4 is a flow chart of a co-prime matrix robust adaptive beamforming algorithm based on matrix filling.
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It is to be noted that the features in the following embodiments and examples may be combined with each other without conflict.
It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention, and the components related to the present invention are only shown in the drawings rather than drawn according to the number, shape and size of the components in actual implementation, and the type, quantity and proportion of the components in actual implementation may be changed freely, and the layout of the components may be more complicated.
As shown in fig. 4, the present invention provides a matrix filling based co-prime array robust adaptive beamforming algorithm, where the co-prime array includes a first ULA and a second ULA, and the first ULA and the second ULA are co-prime; the algorithm comprises the following steps:
s1 calculating a sample covariance matrix of received data
Figure GDA0003083536180000061
S2 covariance matrix of the samples
Figure GDA0003083536180000062
Vectorizing to obtain a vector z, and then performing redundancy removal and vector rearrangement on the vector z to obtain a received data vector z of the complete co-prime matrix differential optimization array1；
S3 is to receive the data vector z1All the elements with discontinuous medium wave path difference are filled with 0 to obtain vector
Figure GDA0003083536180000063
Then taking the vector
Figure GDA0003083536180000064
To obtain a vector
Figure GDA0003083536180000065
S4 fitting the vector
Figure GDA0003083536180000066
Expanded into Topritz matrices Rv∈C(2MN-N+1)×(2MN-N+1)Wherein 2M is the number of array elements of the first ULA, N is the number of array elements of the second ULA, M, N are mutually prime numbers, and M is<N；
S5 fitting the Topritz matrix RvRecovering to obtain a filled covariance matrix
Figure GDA0003083536180000067
S6 fitting the filling covariance matrix
Figure GDA0003083536180000068
Performing characteristic decomposition and utilizationFilling covariance matrices
Figure GDA0003083536180000069
Noise subspace U ofnConstructing a MUSIC spectrum; in the desired signal angle region ΘsInternally performing spectral peak search to obtain the arrival angle of the expected signal
Figure GDA00030835361800000610
Calculating a steering vector of the desired signal
Figure GDA00030835361800000611
In the angular region of the interference signal
Figure GDA00030835361800000612
The estimation of the arrival angle of each interference signal is obtained by searching the spectrum peak in the interior
Figure GDA00030835361800000613
S7, using the estimated arrival angle of the interference signal and the physical array information of the co-prime matrix and reconstructing the interference-plus-noise covariance matrix of the co-prime matrix physical array
Figure GDA00030835361800000614
S8 using the interference plus noise covariance matrix
Figure GDA00030835361800000615
And estimation of a desired signal steering vector
Figure GDA00030835361800000616
A weight vector w for the adaptive beamformer is calculated.
In one embodiment, in step S1, a sample covariance matrix is calculated using K sample snapshots:
Figure GDA00030835361800000617
wherein: k is the number of sampling times, K is the serial number of sampling, x (K) is the data received by each array element of the co-prime array, xH(k) Is the conjugate transpose of x (k).
In an embodiment, in step S2, the performing redundancy removal and vector rearrangement on the vector z specifically includes:
the sums of the elements with the same path difference are averaged as follows:
Figure GDA0003083536180000071
wherein m represents the wave path difference of the difference optimization array, | · | represents the number of elements in the solution set,<x>irepresenting the value of a signal x at the element reference position i,
Figure GDA0003083536180000072
representing the value of the covariance matrix R corresponding to the wave path difference i-j;
Figure GDA0003083536180000073
representing a covariance matrix
Figure GDA0003083536180000074
Corresponding wave path difference of n1-n2The value of (a) is (b),<x>n1representing a signal vector x at a reference position n of an array element1The value of (a) is (b),
Figure GDA0003083536180000075
representing a signal vector x at a reference position n of an array element2The value of (a) is (b),
set T (m) represents a set of array element position doublets with wave path difference m
T(m)＝{(n1,n2)∈S2|n1-n2＝m}
Will be provided with<z1>mArranging according to the order of the wave path difference from small to large to obtain the receiving data vector z of the corresponding complete co-prime matrix differential optimization array1∈C3MN+M-N。
In one embodiment, in step S3, the receiving data vector z is1All the elements with discontinuous medium wave path difference are filled with 0 to obtain vector
Figure GDA0003083536180000076
Then taking the vector
Figure GDA0003083536180000077
To obtain a vector
Figure GDA0003083536180000078
The method specifically comprises the following steps:
the received data vector z1All the elements with discontinuous middle wave path difference are filled with 0, so that they form a higher-dimensional received data vector
Figure GDA0003083536180000079
Namely, it is
Figure GDA00030835361800000710
Wherein S isdiffSet of values representing positions of elements of a differentially optimised array, Sdiff＝{n1-n2|n1,n2∈S}，S＝{dnD, N is 1, 2M + N-1, d represents a half-wave long distance λ/2;
taking the received data vector
Figure GDA00030835361800000711
The positive half of the vector data is obtained to obtain a complex vector with the dimensionality of 2MN-N +1
Figure GDA00030835361800000712
Figure GDA00030835361800000713
In one embodiment, in step S5, a matrix filling problem is constructed from the Topritz matrix RvRecovering to obtain a filled covariance matrix
Figure GDA00030835361800000714
Figure GDA0003083536180000081
Figure GDA0003083536180000082
Figure GDA0003083536180000083
n1,n2∈S+＝{n|n∈S,n≥0}
Wherein rank () represents the rank of the matrix, and the rank minimization problem is converted into the kernel norm minimization problem as follows
Figure GDA0003083536180000084
Figure GDA0003083536180000085
Figure GDA0003083536180000086
n1,n2∈S+
Wherein | · | purple sweet*Representing the kernel norm of the matrix.
In one embodiment, in step S6, the estimation of the angle of arrival of each interference signal
Figure GDA0003083536180000087
Obtained by the following method:
for the filled covariance matrix
Figure GDA0003083536180000088
The characteristic decomposition is carried out, and the characteristic decomposition is carried out,
Figure GDA0003083536180000089
wherein, UsIs composed of
Figure GDA00030835361800000810
Signal subspace of, sigmasIs UsThe characteristic value, U, corresponding to each vectornIs composed of
Figure GDA00030835361800000811
Of noise subspace, ΣnIs UnThe characteristic value corresponding to each vector in the vector;
using the noise subspace UnConstructing a MUSIC spectrum on the whole angle area,
Figure GDA00030835361800000812
wherein θ ∈ [ -90 °,90 °]Representing angles in the search, d (θ) representing a filled covariance matrix
Figure GDA00030835361800000813
Corresponding to the steering vector at the angle θ, d (θ) < 1, ej2πdsinθλ，ej2π2dsinθλθ/λ...，ej2π(2MN-N)dsinθλ]T；
Assuming that the desired signal is located in a span
Figure GDA00030835361800000815
Wherein
Figure GDA00030835361800000816
Is a presumed expectationAngle of arrival of signal, interference signal is located at thetasBetween the supplement section
Figure GDA00030835361800000817
Performing the following steps;
in the desired signal interval thetasInternal pair MUSIC spectrum PMUSIC(theta) performing a spectral peak search, using the largest spectral peak in the region as the spectral peak of the desired signal, and determining the angular position thereof
Figure GDA00030835361800000818
Recording an estimate of the angle of arrival as a desired signal;
the estimation of the steering vector of the desired signal is as follows;
Figure GDA0003083536180000091
in the interval of interference signal
Figure GDA0003083536180000092
The arrival angle information of each interference signal is obtained by searching the spectrum peak of the MUSIC spectrum, and the estimation of the arrival angle of each interference signal is obtained after the largest (L-1) spectrum peak is selected
Figure GDA0003083536180000093
In one embodiment, in step S7, the interference plus noise covariance matrix
Figure GDA0003083536180000094
In order to realize the purpose,
Figure GDA0003083536180000095
wherein, γminRepresenting a sample covariance matrix
Figure GDA0003083536180000096
Is determined by the minimum characteristic value of (c),
Figure GDA0003083536180000097
estimated power for the l source, I2M+N-1The dimension is a unit array of 2MN + N-1;
in one embodiment, in step S8, the weighting vector w of the adaptive beamformer is:
Figure GDA0003083536180000098
according to the invention, through a matrix filling technology, the holes of the discontinuous part of the differential optimization array of the co-prime array are interpolated, so that the whole differential optimization array forms a continuous ULA, the data of all virtual array elements of the differential optimization array are effectively utilized, and the degree of freedom of the differential optimization array of the co-prime array, which can be actually utilized by an algorithm, is improved. In the adaptive beamforming technology, the higher the degree of freedom of an array for processing a received signal is, the more accurately an interference signal can be located and suppressed, thereby obtaining better beamforming performance.
The invention is described below with specific data,
a mutual prime matrix robust adaptive beamforming algorithm based on matrix filling comprises the following steps:
step 11, setting an antenna array:
the antenna array is a co-prime array constructed from a pair of ULA (regular linear array) with a co-prime spacing between the array elements. Determining a pair of prime numbers M-3, N-5, M<And N is added. The first ULA is composed of 6 array elements with Nd-5 d spacing, 2M, and the second ULA is composed of 5 array elements with Md-3 d spacing, where d represents half-wave long distance λ/2, and in this embodiment, the signal wavelength λ is 0.375M, so d is 0.1875M. The final co-prime array has (2M + N-1 ═ 10) array elements. The existing L-4 far-field narrow-band signals are incident on a co-prime array, and DOA of the signals are { theta [ ]1＝0°,θ2＝-30°,θ3＝30°,θ445 deg., DOA of the desired signal is θ1And the other DOAs are the arrival angles of the interference signals.
Step 12: computing a sample covariance matrix for received data
Figure GDA0003083536180000099
The sample covariance matrix is calculated using 500 sample snapshots as follows:
Figure GDA00030835361800000910
wherein: k is the serial number of sampling, and x (k) is the data received by each array element of the co-prime array and is arranged in a line according to the sequence of the array elements.
Next, the covariance matrix of the sample is calculated
Figure GDA0003083536180000101
Performing characteristic decomposition to calculate the minimum characteristic value gammaminFor subsequent calculations.
Step 13: covariance matrix of samples
Figure GDA0003083536180000102
Vectorization results in a vector z that is,
Figure GDA0003083536180000103
carrying out redundancy removal and vector rearrangement on the vector z according to the corresponding wave path difference to obtain a received data vector z of the complete co-prime matrix differential optimization array1. The specific operation is as follows:
when de-redundancy is performed for elements of the same path difference, the elements are summed and averaged, i.e. the sum is averaged
Figure GDA0003083536180000104
Will be provided with<z1>mArranging according to the order of the wave path difference from small to large, and obtaining the receiving data vector z of the corresponding complete co-prime matrix differential optimization array1∈C43。
Step 14: will receive a data vector z1All the elements with discontinuous middle wave path difference are filled with 0, so that they form a higher-dimensional received data vector
Figure GDA0003083536180000105
Namely, it is
Figure GDA0003083536180000106
Taking received data vectors
Figure GDA0003083536180000107
To obtain a vector
Figure GDA0003083536180000108
Figure GDA0003083536180000109
Step 15: according to the characteristic of conjugate symmetry of signal covariance matrix, vector is converted into linear vector
Figure GDA00030835361800001010
Expanded into Topritz matrices Rv∈C26×26，
Figure GDA00030835361800001011
Step 16: toplitz matrix R due to the low rank nature of the received data covariance matrixvThe 0 element in (1) can be effectively recovered by constructing the following kernel norm minimization problem, thereby obtaining a filled covariance matrix
Figure GDA00030835361800001012
Figure GDA00030835361800001013
Figure GDA00030835361800001014
Figure GDA00030835361800001015
n1,n2∈S+
The nuclear norm minimization problem is a convex optimization problem, can be effectively solved through semi-definite programming, and can be conveniently and directly solved by using a convex optimization tool kit CVX.
And step 17: for the filled covariance matrix
Figure GDA0003083536180000111
Performing feature decomposition by using its noise subspace UnAnd constructing a MUSIC spectrum. In the desired signal angle region ΘsInternally performing spectral peak search to obtain the arrival angle of the expected signal
Figure GDA0003083536180000112
Calculating a steering vector of the desired signal
Figure GDA0003083536180000113
In the angular region of the interference signal
Figure GDA0003083536180000114
The estimation of the arrival angle of each interference signal is obtained by searching the spectrum peak in the interior
Figure GDA0003083536180000115
The method comprises the following specific steps:
step 1701: for the filled covariance matrix
Figure GDA0003083536180000116
The characteristic decomposition is carried out, and the characteristic decomposition is carried out,
Figure GDA0003083536180000117
wherein, UsBy
Figure GDA0003083536180000118
Is composed of the main feature vectors of
Figure GDA0003083536180000119
Signal subspace of, sigmasIs UsThe characteristic value corresponding to each vector in the vector; u shapenBy
Figure GDA00030835361800001110
Is composed of small feature vectors of
Figure GDA00030835361800001111
Of noise subspace, ΣnIs UnThe corresponding eigenvalue of each vector.
Step 1702: using noise subspaces UnConstructing a MUSIC spectrum on the whole angle area,
Figure GDA00030835361800001112
wherein θ ∈ [ -90 °,90 °]Representing angles in the search, d (θ) representing a filled covariance matrix
Figure GDA00030835361800001113
Corresponding to the steering vector at the angle θ, d (θ) < 1, ejπsinθ,ej2πsinθ,...,ej25πsinθ]T。
Step 1703: in this embodiment, Θs＝[-5°,5°]The interference signal is located at thetasBetween the supplement section
Figure GDA00030835361800001114
In (1).
First, in the desired signal interval ΘsInternal pair MUSIC spectrum PMUSIC(theta) performing a spectral peak search, using the largest spectral peak in the region as the spectral peak of the desired signal, and determining the angular position thereof
Figure GDA00030835361800001115
Recording an estimate of the angle of arrival as a desired signal; at theta, when the input signal-to-noise ratio is lowsIt is often difficult to search for a spectral peak of the desired signal, and then the assumed angle of the desired signal is used as an estimate of the angle of arrival of the desired signal, and the steering vector of the desired signal is calculated as follows.
Figure GDA00030835361800001116
Then, in the interference signal interval
Figure GDA00030835361800001117
And carrying out spectrum peak search on the MUSIC spectrum to obtain the arrival angle information of each interference signal. After the largest 3 spectral peaks are selected, the estimation of the arrival angle of each interference signal can be obtained
Figure GDA00030835361800001118
Step 18: order to
Figure GDA00030835361800001119
Representing the set of estimated individual source DOAs. Next, using θmThe following least squares problem can be constructed
Figure GDA00030835361800001120
subject to p(θm)＞0,
Wherein the content of the first and second substances,
Figure GDA00030835361800001121
representative set θmThe distribution of the power over the power-supply line,
Figure GDA00030835361800001122
the estimated power for the ith source is the target for the solution of the optimization problem. diag {. represents the diagonalization operation of the vector.
Figure GDA0003083536180000121
Is thetamCorresponding array flow pattern. Estimation of noise power
Figure GDA0003083536180000122
Taken as a sample covariance matrix
Figure GDA0003083536180000123
The minimum eigenvalue of (c). In fact, when solving the optimization problem, inequality constraints can be ignored, so as to obtain the following least square closed-form solution
p(θm)＝(GHG)-1GHr
Wherein the content of the first and second substances,
Figure GDA0003083536180000124
step 19: using the estimated angle of arrival of the interfering signal
Figure GDA0003083536180000125
And the physical array information of the co-prime array, and reconstructing the interference and noise covariance matrix of the co-prime array physical array
Figure GDA0003083536180000126
Figure GDA0003083536180000127
Step 110: using interference plus noise covariance matrices
Figure GDA0003083536180000128
And estimation of a desired signal steering vector
Figure GDA0003083536180000129
Calculating a weight vector w of the adaptive beamformer:
Figure GDA00030835361800001210
to further verify the effectiveness of the above embodiments, a simulation experiment was designed in which the desired signal steering vector was accurately known in consideration. In the simulation experiment, the additive noise is complex gaussian zero mean white noise and has the same variance at each array element. The sample data always contains a desired signal component. The interference signal has a dry-to-noise ratio INR equal to 30dB at each sensor. In an experiment for analyzing the change of the output signal-to-interference-and-noise ratio along with the snapshot number, the input signal-to-noise ratio is fixed to be 20 dB; in the experiment for analyzing the change of the output signal-to-interference-and-noise ratio along with the input signal-to-noise ratio, the snapshot number is fixed to be K-500. To obtain each data point in each experiment, 500 monte carlo experiments were performed.
Fig. 2 depicts a plot of output SINR of the tested beamformer as a function of input SNR, and fig. 3 is a plot of output SINR of the tested beamformer as a function of sample snapshot. It can be seen that the performance of the proposed algorithm is still better than other comparison algorithms, and the convergence speed is faster than other comparison algorithms.
The foregoing embodiments are merely illustrative of the principles and utilities of the present invention and are not intended to limit the invention. Any person skilled in the art can modify or change the above-mentioned embodiments without departing from the spirit and scope of the present invention. Accordingly, it is intended that all equivalent modifications or changes which can be made by those skilled in the art without departing from the spirit and technical spirit of the present invention be covered by the claims of the present invention.

Claims (7)

1. A co-prime array robust adaptive beamforming algorithm based on matrix filling, wherein the co-prime array comprises a first ULA and a second ULA, said first ULA being co-prime with said second ULA; the algorithm comprises the following steps:
computing a sample covariance matrix for received data
Figure FDA0003370289740000011
The sample covariance matrix
Figure FDA0003370289740000012
Vectorizing to obtain a vector z, and then performing redundancy removal and vector rearrangement on the vector z to obtain a received data vector z of the complete co-prime matrix differential optimization array1；
The received data vector z1All the elements with discontinuous medium wave path difference are filled with 0 to obtain vector
Figure FDA0003370289740000013
Then taking the vector
Figure FDA0003370289740000014
To obtain a vector
Figure FDA0003370289740000015
The vector is measured
Figure FDA0003370289740000016
Expanded into Topritz matrices Rv∈C(2MN-N+1)×(2MN-N+1)Wherein 2M is the number of array elements of the first ULA, N is the number of array elements of the second ULA, M, N are mutually prime numbers, and M is<N；
Subjecting the Topritz matrix RvRecovering to obtain a filled covariance matrix
Figure FDA0003370289740000017
For the filled covariance matrix
Figure FDA0003370289740000018
Performing feature decomposition by using the filled covariance matrix
Figure FDA0003370289740000019
Noise subspace U ofnConstructing a MUSIC spectrum; in the desired signal angle region ΘsInternally performing spectral peak search to obtain the arrival angle of the expected signal
Figure FDA00033702897400000110
Calculating a steering vector of the desired signal
Figure FDA00033702897400000111
In the angular region of the interference signal
Figure FDA00033702897400000112
The estimation of the arrival angle of each interference signal is obtained by searching the spectrum peak in the interior
Figure FDA00033702897400000113
Using the estimated arrival angle of the interference signal and the physical array information of the co-prime matrix and reconstructing the interference-plus-noise covariance matrix of the co-prime matrix physical array
Figure FDA00033702897400000114
Using the interference-plus-noise covariance matrix
Figure FDA00033702897400000115
And estimation of a desired signal steering vector
Figure FDA00033702897400000116
Calculating a weighting vector w of the adaptive beam former;
the interference plus noise covariance matrix
Figure FDA00033702897400000117
In order to realize the purpose,
Figure FDA00033702897400000118
wherein, γminRepresenting a sample covariance matrix
Figure FDA00033702897400000119
Is determined by the minimum characteristic value of (c),
Figure FDA00033702897400000120
estimated power for the l source, I2M+N-1Is a unit array with the dimension of 2MN + N-1.
2. The matrix filling-based mutual-prime-array robust adaptive beamforming algorithm according to claim 1, wherein the sample covariance matrix is calculated using K sample snapshots:
Figure FDA00033702897400000121
wherein: k is the number of sampling times, K is the serial number of sampling, x (K) is the data received by each array element of the co-prime array, xH(k) Is the conjugate transpose of x (k).
3. The algorithm according to claim 2, wherein the performing redundancy removal and vector rearrangement on the vector z specifically comprises:
the sums of the elements with the same path difference are averaged as calculated below
Figure FDA0003370289740000021
Wherein m represents the wave path difference of the difference optimization array, | · | represents the number of elements in the solution set,
Figure FDA0003370289740000022
representing a covariance matrix
Figure FDA0003370289740000023
Corresponding wave path difference of n1-n2The value of (a) is (b),
Figure FDA0003370289740000024
representing a signal vector x at a reference position n of an array element1The value of (a) is (b),
Figure FDA0003370289740000025
representing a signal vector x at a reference position n of an array element2The value of (d);
set T (m) represents a set of array element position doublets with wave path difference m
T(m)＝{(n1,n2)∈S2|n1-n2＝m}
Will be provided with<z1>mArranging according to the order of the wave path difference from small to large to obtain the receiving data vector z of the corresponding complete co-prime matrix differential optimization array1∈C3MN+M-N。
4. The matrix filling-based co-prime matrix robust adaptive beamforming algorithm according to claim 3, wherein the received data vector z1All the elements with discontinuous medium wave path difference are filled with 0 to obtain vector
Figure FDA0003370289740000026
Then taking the vector
Figure FDA0003370289740000027
To obtain a vector
Figure FDA0003370289740000028
The method specifically comprises the following steps:
the received data vector z1All the elements with discontinuous middle wave path difference are filled with 0, so that they form a higher-dimensional received data vector
Figure FDA0003370289740000029
Namely, it is
Figure FDA00033702897400000210
Wherein S isdiffSet of values representing positions of elements of a differentially optimised array, Sdiff＝{n1-n2|n1,n2∈S}，S＝{dnD, N is 1, 2M + N-1, d represents a half-wave long distance λ/2;
taking the received data vector
Figure FDA00033702897400000211
The positive half of the vector data is obtained to obtain a complex vector with the dimensionality of 2MN-N +1
Figure FDA00033702897400000212
5. The algorithm of claim 4, wherein the matrix filling problem is constructed to be solved by the Toeplitz matrix RvRecovering to obtain a filled covariance matrix
Figure FDA00033702897400000213
Figure FDA0003370289740000031
Figure FDA0003370289740000032
Figure FDA0003370289740000033
n1,n2∈S+＝{n|n∈S,n≥0}
Wherein rank () represents the rank of the matrix, and the rank minimization problem is converted into the kernel norm minimization problem as follows
Figure FDA0003370289740000034
Figure FDA0003370289740000035
Figure FDA0003370289740000036
n1,n2∈S+
Wherein | · | purple sweet*Representing the kernel norm of the matrix.
6. The matrix-filling based mutual-prime-array robust adaptive beamforming algorithm according to claim 5, wherein the estimation of the angle of arrival of each interference signal
Figure FDA0003370289740000037
Obtained by the following method:
for the filled covariance matrix
Figure FDA0003370289740000038
The characteristic decomposition is carried out, and the characteristic decomposition is carried out,
Figure FDA0003370289740000039
wherein, UsIs composed of
Figure FDA00033702897400000310
Signal subspace of, sigmasIs UsThe characteristic value, U, corresponding to each vectornIs composed of
Figure FDA00033702897400000311
Of noise subspace, ΣnIs UnThe characteristic value corresponding to each vector in the vector;
using the noise subspace UnConstructing a MUSIC spectrum on the whole angle area,
Figure FDA00033702897400000312
wherein θ ∈ [ -90 °,90 °]Representing angles in the search, d (θ) representing a filled covariance matrix
Figure FDA00033702897400000313
Corresponding to the steering vector at the angle θ, d (θ) < 1, ej2πdsinθ/λ,ej2π2dsinθ/λ...,ej2π(2MN-N)dsinθ/λ]T；
Assuming that the desired signal is located in a span
Figure FDA00033702897400000314
Wherein
Figure FDA00033702897400000315
Is the angle of arrival of the assumed desired signal, and the interfering signal is located at thetasBetween the supplement section
Figure FDA00033702897400000316
Performing the following steps;
in the desired signal interval thetasInternal pair MUSIC spectrum PMUSIC(theta) performing a spectral peak search to find the region of the maximaThe large spectral peak is taken as the spectral peak of the desired signal, and the angular position is determined
Figure FDA00033702897400000317
Recording an estimate of the angle of arrival as a desired signal;
the estimation of the steering vector of the desired signal is as follows;
Figure FDA00033702897400000318
in the interval of interference signal
Figure FDA0003370289740000041
The arrival angle information of each interference signal is obtained by searching the spectrum peak of the MUSIC spectrum, and the estimation of the arrival angle of each interference signal is obtained after the largest (L-1) spectrum peak is selected
Figure FDA0003370289740000042
7. The algorithm according to claim 1, wherein the weighting vector w of the adaptive beamformer is:
Figure FDA0003370289740000043
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